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%I M4225 N1766 #172 Nov 22 2023 15:33:14
%S 1,6,36,240,1800,15120,141120,1451520,16329600,199584000,2634508800,
%T 37362124800,566658892800,9153720576000,156920924160000,
%U 2845499424768000,54420176498688000,1094805903679488000,23112569077678080000,510909421717094400000
%N Lah numbers: a(n) = (n-1)*n!/2.
%C Number of surjections from {1,...,n} to {1,...,n-1}. - _Benoit Cloitre_, Dec 05 2003
%C First Eulerian transform of 0,1,2,3,4,... . - _Ross La Haye_, Mar 05 2005
%C With offset 0 : determinant of the n X n matrix m(i,j)=(i+j+1)!/i!/j!. - _Benoit Cloitre_, Apr 11 2005
%C These numbers arise when expressing n(n+1)(n+2)...(n+k)[n+(n+1)+(n+2)+...+(n+k)] as sums of squares: n(n+1)[n+(n+1)] = 6(1+4+9+16+ ... + n^2), n(n+1)(n+2)(n+(n+1)+(n+2)) = 36(1+(1+4)+(1+4+9)+...+(1+4+9+16+ ... + n^2)), n(n+1)(n+2)(n+3)(n+(n+1)+(n+2)+(n+3)) = 240(...), ... . - _Alexander R. Povolotsky_, Oct 16 2006
%C a(n) is the number of edges in the Hasse diagram for the weak Bruhat order on the symmetric group S_n. For permutations p,q in S_n, q covers p in the weak Bruhat order if p,q differ by an adjacent transposition and q has one more inversion than p. Thus 23514 covers 23154 due to the transposition that interchanges the third and fourth entries. Cf. A002538 for the strong Bruhat order. - _David Callan_, Nov 29 2007
%C a(n) is also the number of excedances in all permutations of {1,2,...,n} (an excedance of a permutation p is a value j such p(j)>j). Proof: j is exceeded (n-1)! times by each of the numbers j+1, j+2, ..., n; now, Sum_{j=1..n} (n-j)(n-1)! = n!(n-1)/2. Example: a(3)=6 because the number of excedances of the permutations 123, 132, 312, 213, 231, 321 are 0, 1, 1, 1, 2, 1, respectively. - _Emeric Deutsch_, Dec 15 2008
%C (-1)^(n+1)*a(n) is the determinant of the n X n matrix whose (i,j)-th element is 0 for i = j, is j-1 for j>i, and j for j < i. - _Michel Lagneau_, May 04 2010
%C Row sums of the triangle in A030298. - _Reinhard Zumkeller_, Mar 29 2012
%C a(n) is the total number of ascents (descents) over all n-permutations. a(n) = Sum_{k=1..n} A008292(n,k)*k. - _Geoffrey Critzer_, Jan 06 2013
%C For m>=4, a(m-2) is the number of Hamiltonian cycles in a simple graph with m vertices which is complete, except for one edge. Proof: think of distinct round-table seatings of m persons such that persons "1" and "2" may not be neighbors; the count is (m-3)(m-2)!/2. See also A001710. - _Stanislav Sykora_, Jun 17 2014
%C Popularity of left (right) children in treeshelves. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n. See A278677, A278678 or A278679 for more definitions and examples. See A008292 for the distribution of the left (right) children in treeshelves. - _Sergey Kirgizov_, Dec 24 2016
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 90, ex. 4.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%D John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001286/b001286.txt">Table of n, a(n) for n = 2..100</a>
%H Yasmin Aguillon et al., <a href="https://arxiv.org/abs/2206.00541">On Parking Functions and The Tower of Hanoi</a>, arXiv:2206.00541 [math.CO], 2022.
%H Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1611.07793">Patterns in treeshelves</a>, arXiv:1611.07793 [cs.DM], 2016.
%H Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, <a href="https://arxiv.org/abs/2302.08265">MC-finiteness of restricted set partition functions</a>, arXiv:2302.08265 [math.CO], 2023.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=399">Encyclopedia of Combinatorial Structures 399</a>.
%H Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, <a href="https://arxiv.org/abs/2306.14734">Boolean intervals in the weak order of S_n</a>, arXiv:2306.14734 [math.CO], 2023.
%H Lucas Chaves Meyles, Pamela E. Harris, Richter Jordaan, Gordon Rojas Kirby, Sam Sehayek, and Ethan Spingarn, <a href="https://arxiv.org/abs/2305.15554">Unit-Interval Parking Functions and the Permutohedron</a>, arXiv:2305.15554 [math.CO], 2023.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Sandi Klavžar, Uroš Milutinović and Ciril Petr, <a href="http://dx.doi.org/10.1016/j.exmath.2005.05.003">Hanoi graphs and some classical numbers</a>, Expo. Math. 23 (2005), no. 4, 371-378.
%H S. Lehr, J. Shallit and J. Tromp, <a href="http://dx.doi.org/10.1016/0304-3975(95)00234-0">On the vector space of the automatic reals</a>, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
%H P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BruhatGraph.html">Bruhat Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationAscent.html">Permutation Ascent</a>.
%F a(n) = Sum_{i=0..n-1} (-1)^(n-i-1) * i^n * binomial(n-1,i). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000 [corrected by _Amiram Eldar_, May 02 2022]
%F E.g.f.: x^2/[2(1-x)^2]. - _Ralf Stephan_, Apr 02 2004
%F a(n+1) = (-1)^(n+1)*det(M_n) where M_n is the n X n matrix M_(i,j)=max(i*(i+1)/2,j*(j+1)/2). - _Benoit Cloitre_, Apr 03 2004
%F Row sums of table A051683. - _Alford Arnold_, Sep 29 2006
%F 5th binomial transform of A135218: (1, 1, 1, 25, 25, 745, 3145, ...). - _Gary W. Adamson_, Nov 23 2007
%F If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n)=(-1)^n*f(n,2,-2), (n>=2). - _Milan Janjic_, Mar 01 2009
%F a(n) = A000217(n-1)*A000142(n-1). - _Reinhard Zumkeller_, May 15 2010
%F a(n) = (n+1)!*Sum_{k=1..n-1} 1/(k^2+3*k+2). - _Gary Detlefs_, Sep 14 2011
%F Sum_{n>=2} 1/a(n) = 2*(2 - exp(1) - gamma + Ei(1)) = 1.19924064599..., where gamma = A001620 and Ei(1) = A091725. - _Ilya Gutkovskiy_, Nov 24 2016
%F a(n+1) = a(n)*n*(n+1)/(n-1). - _Chai Wah Wu_, Apr 11 2018
%F Sum_{n>=2} (-1)^n/a(n) = 2*(gamma - Ei(-1)) - 2/e, where e = A001113 and Ei(-1) = -A099285. - _Amiram Eldar_, May 02 2022
%e G.f. = x^2 + 6*x^3 + 36*x^4 + 240*x^5 + 1800*x^6 + 15120*x^7 + 141120*x^8 + ...
%e a(10) = (1+2+3+4+5+6+7+8+9)*(1*2*3*4*5*6*7*8*9) = 16329600. - _Reinhard Zumkeller_, May 15 2010
%p seq(sum(mul(j,j=3..n), k=2..n), n=2..21); # _Zerinvary Lajos_, Jun 01 2007
%t Table[Sum[n!, {i, 2, n}]/2, {n, 2, 20}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t nn=20;With[{a=Accumulate[Range[nn]],t=Range[nn]!},Times@@@Thread[{a,t}]] (* _Harvey P. Dale_, Jan 26 2013 *)
%t Table[(n - 1) n! / 2, {n, 2, 30}] (* _Vincenzo Librandi_, Sep 09 2016 *)
%o (Sage) [(n-1)*factorial(n)/2 for n in range(2, 21)] # _Zerinvary Lajos_, May 16 2009
%o (Haskell)
%o a001286 n = sum[1..n-1] * product [1..n-1]
%o -- _Reinhard Zumkeller_, Aug 01 2011
%o (Maxima) A001286(n):=(n-1)*n!/2$
%o makelist(A001286(n),n,1,30); /* _Martin Ettl_, Nov 03 2012 */
%o (PARI) a(n)=(n-1)*n!/2 \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [(n-1)*Factorial(n)/2: n in [2..25]]; // _Vincenzo Librandi_, Sep 09 2016
%o (Python)
%o from __future__ import division
%o A001286_list = [1]
%o for n in range(2,100):
%o A001286_list.append(A001286_list[-1]*n*(n+1)//(n-1)) # _Chai Wah Wu_, Apr 11 2018
%Y Cf. A001710, A052609, A062119, A075181, A060638, A060608, A060570, A060612, A135218, A019538, A053495, A051683, A213168, A278677, A278678, A278679, A008292.
%Y A002868 is an essentially identical sequence.
%Y Column 2 of |A008297|.
%Y Third column (m=2) of triangle |A111596(n, m)|: matrix product of |S1|.S2 Stirling number matrices.
%Y Cf. also A000110, A000111.
%Y Cf. A001113, A001620, A091725, A099285.
%K nonn,easy,nice
%O 2,2
%A _N. J. A. Sloane_
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